Domains of Positivity

A Domain of Positivity D is an open convex cone associated with a nonsingular symmetric matrix S, called the characteristic, such that xÇzD if and only if x'Sy>0 for all y(~D* As such they were introduced by Koecher (1) in generalization of the cone of positive definite matrices studied by Siegel. The automorphisms of D are the nonsingular linear transformations mapping D onto itself. The group of automorphisms {W] admits an anti-automorphism: W—>S~W'Sf where W' means W transposed. A norm N(x) is a function positive and continuous for x£J9 and satisfying there N(Wx) =||TF||N(X) for every automorphism W. A norm is given by: